The Volume of Prisms and Cylinders and Cavalieri’s Principle

New York State Common Core Math Geometry, Module 3, Lesson 10

Worksheets for Geometry

Student Outcomes

• Students understand the principle of parallel slices in the plane, and understand Cavalieri’s principle as a generalization of the principle of parallel slices.
• Students use Cavalieri’s principle to reason that the volume formula for a general cylinder is area of base×height.

The Volume of Prisms and Cylinders and Cavalieri’s Principle

Classwork

Opening Exercise

The bases of the following triangular prism 𝑇 and rectangular prism 𝑅 lie in the same plane. A plane that is parallel to the bases and also a distance 3 from the bottom base intersects both solids and creates cross-sections 𝑇′ and 𝑅′. a. Find Area(𝑇′). b. Find Area(𝑅′). c. Find Vol(𝑇). d. Find Vol(𝑅). e. If a height other than 3 were chosen for the cross-section, would the cross-sectional area of either solid change

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Discussion

PRINCIPLE OF PARALLEL SLICES IN THE PLANE: If two planar figures of equal altitude have identical cross-sectional lengths at each height, then the regions of the figures have the same area.

Example

a. The following triangles have equal areas: Area(△ 𝐴𝐵𝐶) = Area(△ 𝐴′𝐵′𝐶′ = 15 units². The distance between 𝐷𝐸 and 𝐶𝐶′is 3. Find the lengths of 𝐷𝐸 and 𝐷′𝐸′. b. Joey says that if two figures have the same height and the same area, then their cross-sectional lengths at each height will be the same. Give an example to show that Joey’s theory is incorrect.

Discussion

CAVALIERI’S PRINCIPLE: Given two solids that are included between two parallel planes, if every plane parallel to the two planes intersects both solids in cross-sections of equal area, then the volumes of the two solids are equal

Lesson Summary

PRINCIPLE OF PARALLEL SLICES IN THE PLANE: If two planar figures of equal altitude have identical cross-sectional lengths at each height, then the regions of the figures have the same area.

CAVALIERI’S PRINCIPLE: Given two solids that are included between two parallel planes, if every plane parallel to the two planes intersects both solids in cross-sections of equal area, then the volumes of the two solids are equal.

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